Multipoint Fractional Iterative Methods with (2<i>α</i> + 1)th-Order of Convergence for Solving Nonlinear Problems

• Main Authors: Giro Candelario, Alicia Cordero, Juan R. Torregrosa
• Format: Article
• Language: English
• Published: MDPI AG 2020-03-01
• Series: Mathematics
• Subjects: nonlinear equations; fractional derivatives; multistep methods; convergence; stability
• Online Access: https://www.mdpi.com/2227-7390/8/3/452

In the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#945;</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and compare it with the existing fractional Newton method with order <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>&#945;</mi> </mrow> </semantics> </math> </inline-formula>. Moreover, we also introduce a multipoint fractional Traub-type method with order <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>&#945;</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and compare its performance with that of its first step. Some numerical tests and analysis of the dependence on the initial estimations are made for each case, including a comparison with classical Newton (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#945;</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> of the first step of the class) and classical Traub&#8217;s scheme (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#945;</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> of fractional proposed multipoint method). In this comparison, some cases are found where classical Newton and Traub&#8217;s methods do not converge and the proposed methods do, among other advantages.